Empty convex polygons in planar point sets
نویسنده
چکیده
The Erdős–Szekeres theorem inspired a lot of research. A frequent topic in this area is the study of the existence of so-called empty convex polygons in finite planar point sets. Let P be a finite set of points in general position in the plane. A convex k-gon G is called a k-hole (or empty convex k-gon) of P , if all vertices of G lie in P and no point of P lies inside G. Frequently we will mean by a k-hole the vertex set of G. Erdős [3] asked if, for a fixed k, any sufficiently large point set in general position has a k-hole. Already many years ago, this was known to be true for k ≤ 5 [6] and false for k ≥ 7 [7]. The remaining case k = 6 became a well-known open problem. Gerken [5] (see also [14]) has solved it in the affirmative. Somewhat later the same result was also obtained independently by Nicolás [10]. Thus, we have
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